Verdasco needed two hours and 21 minutes to advance past Dimitrov into the fourth round for the seventh time in his 15th participation in the tournament, reports Efe.
Like her compatriot and good friend Keys, US Open champion Stephens is in the quarter-finals for the first time and hoping to set up a repeat of last year's final in NY.
Even as he conceded the second set, Djokovic managed to regain his composure as he went on to pull off a 6-4 6-7 7-6 6-2 win over Agut in an intriguing third-round battle.
Djokovic had also squandered three set points in the second set and failed to serve out the fourth in the ninth game after being 5-2 in front. He never won a match at a slam, and now maybe he's soon in the semi-finals.
Men's second seed Alexander Zverev survived a scare as he had to save a match point before beating Damir Dzumhur on Friday to reach the fourth round of the French Open.
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Since then, employing a smooth one-handed backhand, he has beaten players seeded No. 8 (David Goffin) and No. 10 (Pablo Carreno Busta), before adding Djokovic to his list.
Djokovic was sent spinning out of the French Open by world number 72 Marco Cecchinato. "After then I warmed up and it was better, but yeah, just a pity that I couldn't capitalise on some break points at 4-1 in the fourth set".
Three-time champion Williams is playing her first Grand Slam tournament since taking time off last year due to pregnancy, while Sharapova is at Roland Garros for the first time in three years after serving a doping ban in 2016 and being refused a wildcard 12 months ago.
After fighting through three consecutive five-set match-wins, Zverev is still on track to be the first German men's champion at Roland Garros since 1937.
"It is the first time in a semi-final in a Grand Slam". I don't know if I'm going to play on grass.
Unseeded Italian Marco Cecchinato caused a major upset at the French Open by knocking out former champion Novak Djokovic to reach the semi-finals.
One break of serve was enough to take the opening set and, despite Verdasco producing some thunderous winners to stay in touch in the second, the left- hander's challenge faded.
A jaw-dropping forehand wrong-footed the Serbian and Cecchinato had a second match point, saved by Djokovic on serve.
Djokovic refused to enter the main press room at Roland Garros after his shock defeat, instead opting to speak to a few journalists in a smaller room where he offered nothing but short answers.